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Unformatted text preview: 328 Finite Element Analysis and Design The problem of a beam under pure bending moment is solved using ﬁve rectangular
ﬁnite elements, as shown in the ﬁgure. Material properties are assumed as: E =
ZOOGPa and v= 0.3. The width of the beam is 0.01 m. In order to simulate the pure
bending moment, two opposite forces F = i 100,000 N are applied at the end of the
beam. Using a commercial FE program, calculate strains of the beam along the
bottom surface. Draw graphs of axx and 7x), as a function of x. Compare the numerical
results with the elementary beam theory results. Provide a theoretical explanation for
the differences. Is the rectangular element stiff or soft? Compared to the CST
element, is the rectangular element stiffer or softer? Normally, a commercial ﬁnite element program provides stress and strain at the corners of the element by averaging the stresses in adjacent elements. Hence, you may use nodal displacement data from FE results to calculate strain along the bottom
surface of the element. Calculate the strains at ten equally spaced points in each
element. Make sure that the commercial program uses the standard Lagrangian shape function Repeat the above procedure when an upward vertical force of 200, 000 N is applied at
the tip of the beam. Use the clamped boundary condition to simulate the cantilevered beam. Solution: In the case of the pure bending problem, the axial stress/strain is constant along the same
ylocation. The beam bending theory yields a 3><10'4 strain along the bottom surface of
the beam, which is in tension. However, the finite element analysis provides 2.02><10‘4
strain, which is much smaller than the exact solution (see Figure (a)). This result shows
that the finite element analysis result is stiffer than exact solution. In addition, according
to beam theory the shear strain for the pure bending problem is zero. However, the ﬁnite element"'analys‘isyiem’s"linearly varying shear strain between :2'><r04 and 23m4 see Figure (bD. Initially perpendicular corner has to be distorted due to the bendingtype
deformation. Such a superﬁcial deformation contributes to the stiff result in the bending problem. However, the rectangular element is softer than triangular element. CHAP 6 Finite Elements for Plane Solids 329 L Bramsauaon 2.8 L
inita Element smipinn With Stmdmrtlﬂane Fumdlm
x \ Rx \ \ Figure (b) Shear strain at the bottom surface As a second example, the cantilevered beam problem is solved using the same ﬁnite elements. Different from the pure bending problem, the axial stress/strain varies linearly
along the beam span. However, as shown in Figure (0), the axial strain varies stepwise
constantly. This is because exx is not a function of xcoordinate but a linear function of y
coordinate. Thus, the axial strain is discontinuous across element boundary. In addition to
that, the averaged axial stress actually has different value from the analytical solution.
This is due to the superﬁcial shear deformation, as shown in Figure (d). Within an
element, the shear strain is a linear function, but discontinuous across element boundary. 330 .05 Finite Element Analysis and Design 0 ‘ \\
0 0.5 1 if} 2 25 '3 3.5 2S 45 5
Figure (0) Axial strain at the bottom surface
x 10:1
2.5 ~ Tla Elvementsmuﬂm Whh Standmd Shane Fumlm "i
x 1 l l I L. _L l i
06 1 $5 2 2‘5 (1 35 4 4.5 6 "Figurefdj Shea? sfrain'at the 'bétt'éin sﬁfface ...
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 Fall '08
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 Finite Element Analysis

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